Families of Orbits in the Vicinity of the Collinear Libration Points
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Families of Orbits in the Vicinity of the Collinear Libration Points K. C. Howell! Abstract In recent years, three-dimensional periodic and quasi -periodic orbits near the collinear libration points in the Sun-Earth/Moon three-body problem have been the focus of great interest for space mission design. Thus, more effort in the astrodynamics community has been directed toward analysis and computation of families of such orbits. But families of periodic orbits in the three-body problem have been the subject of much study for many decades. The first such orbit to be employed for a spacecraft trajectory (ISEE-3) was a member of a particular type of simply symmetric three-dimensional family, i.e., a halo orbit. Thus, attention shifted to further analysis of these periodic halo families. Some of the significant work in the development of periodic solutions in the three-body problem has been reviewed and a number of the highlights from the analysis and eventual numerical computation of halo families is presented here. The halo families of periodic orbits extend from each of the libration points to the nearest primary; they appear to exist for all values of the mass ratio. Thus, this further understanding may serve to support future spacecraft mission planning as well.
Introduction In 1964, Robert Farquhar enrolled at Stanford University where Professor John Breakwell would eventually agree to become his thesis advisor [1]. Near the same time, a challenging problem had captured the interest of Ralph Pringle at Lockheed. In a small study for NASA Marshall, Pringle, who previously collaborated with Breakwell, was investigating spacecraft oscillating about the Earth-Moon L 2 libration point for a communications satellite that could service the backside of the Moon-work that was never published. The suggested solution involved a satellite that would oscillate back and forth in the plane of motion of the Earth and Moon. Such a solution is unsatisfactory because the satellite periodically passes behind the Moon and is not visible from Earth. Farquhar suggested that the problem be addressed by using an out-of-plane solution. The in-plane and out-of-plane frequencies of the motion are close but not exact; the satellite is still out of sight for some time. However, Farquhar's idea incorporated a forced orbit using periodic out-of-plane 'Professor, School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana 47907.
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pulsing [2]. A few months later, Breakwell suggested an alternative continuous controller to force the frequencies to be equal, thus ensuring a periodic out-of-plane solution that is always visible from Earth. This control was slightly cheaper but not very practical [2]. Hence, the basic concept for a "halo orbit" was born. This view of the origin of halo orbits and halo families for space applications is one that is now most familiar to those in the astrodynamics community. And, indeed, this independent insight into bounded motion in the vicinity of the collinear libration point
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